Artificial Neural Networks for Engineers and Scientists: Solving Ordinary Differential Equations
Autor S. Chakraverty, Susmita Mallen Limba Engleză Hardback – 14 iul 2017
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Specificații
ISBN-13: 9781498781381
ISBN-10: 1498781381
Pagini: 168
Ilustrații: 80 Illustrations, black and white
Dimensiuni: 156 x 234 x 14 mm
Greutate: 0.36 kg
Ediția:1
Editura: CRC Press
Colecția CRC Press
ISBN-10: 1498781381
Pagini: 168
Ilustrații: 80 Illustrations, black and white
Dimensiuni: 156 x 234 x 14 mm
Greutate: 0.36 kg
Ediția:1
Editura: CRC Press
Colecția CRC Press
Public țintă
Professional Practice & DevelopmentCuprins
1. Preliminaries of Artificial Neural Network
1.1 Introduction
1.2 Architecture of ANN
1.2.1 Feed-Forward Neural Network
1.2.2 Feedback Neural Network
1.3 Paradigms of Learning
1.3.1 Supervised Learning or Associative Learning
1.3.2 Unsupervised or Self-Organization Learning
1.4 Learning Rules or Learning Processes
1.4.1 Error Back-Propagation Learning Algorithm or Delta
Learning Rule
1.5 Activation Functions
1.5.1 Sigmoid Function
1.5.1.1 Unipolar Sigmoid Function
1.5.1.2 Bipolar Sigmoid Function
1.5.2 Tangent Hyperbolic Function
References
2. Preliminaries of Ordinary Differential Equations
2.1 Definitions
2.1.1 Order and Degree of DEs
2.1.2 Ordinary Differential Equation
2.1.3 Partial Differential Equation
2.1.4 Linear and Nonlinear Differential Equations
2.1.5 Initial Value Problem
2.1.6 Boundary Value Problem
References
3. Multilayer Artificial Neural Network
3.1 Structure of Multilayer ANN Model
3.2 Formulations and Learning Algorithm of Multilayer
ANN Model
3.2.1 General Formulation of ODEs Based on ANN Model
3.2.2 Formulation of nth-Order IVPs
3.2.2.1 Formulation of First-Order IVPs
3.2.2.2 Formulation of Second-Order IVPs
3.2.3 Formulation of BVPs
3.2.3.1 Formulation of Second-Order BVPs
3.2.3.2 Formulation of Fourth-Order BVPs
3.2.4 Formulation of a System of First-Order ODEs
3.2.5 Computation of Gradient of ODEs for Multilayer
ANN Model
3.3 First-Order Linear ODEs
3.4 Higher-Order ODEs
3.5 System of ODEs
References
4. Regression-Based ANN
4.1 Algorithm of RBNN Model
4.2 Structure of RBNN Model
4.3 Formulation and Learning Algorithm of RBNN Model
4.4 Computation of Gradient for RBNN Model
4.5 First-Order Linear ODEs
4.6 Higher-Order Linear ODEs
References
5. Single-Layer Functional Link Artificial Neural Network
5.1 Single-Layer FLANN Models
5.1.1 ChNN Model
5.1.1.1 Structure of the ChNN Model
5.1.1.2 Formulation of the ChNN Model
5.1.1.3 Gradient Computation of the ChNN Model
5.1.2 LeNN Model
5.1.2.1 Structure of the LeNN Model
5.1.2.2 Formulation of the LeNN Model
5.1.2.3 Gradient Computation of the LeNN Model
5.1.3 HeNN Model
5.1.3.1 Architecture of the HeNN Model
5.1.3.2 Formulation of the HeNN Model
5.1.4 Simple Orthogonal Polynomial–Based Neural
Network (SOPNN) Model
5.1.4.1 Structure of the SOPNN Model
5.1.4.2 Formulation of the SOPNN Model
5.1.4.3 Gradient Computation of the SOPNN Model
5.2 First-Order Linear ODEs
5.3 Higher-Order ODEs
5.4 System of ODEs
References
6. Single-Layer Functional Link Artificial Neural Network
with Regression-Based Weights
6.1 ChNN Model with Regression-Based Weights
6.1.1 Structure of the ChNN Model
6.1.2 Formulation and Gradient Computation
of the ChNN Model
6.2 First-Order Linear ODEs
6.3 Higher-Order ODEs
References
7. Lane–Emden Equations
7.1 Multilayer ANN-Based Solution of Lane–Emden Equations
7.2 FLANN-Based Solution of Lane–Emden Equations
7.2.1 Homogeneous Lane–Emden Equations
7.2.2 Nonhomogeneous Lane–Emden Equation
References
8. Emden–Fowler Equations
8.1 Multilayer ANN-Based Solution of Emden–Fowler
Equations
8.2 FLANN-Based Solution of Emden–Fowler Equations
References
9. Duffing Oscillator Equations
9.1 Governing Equation
9.2 Unforced Duffing Oscillator Equations
9.3 Forced Duffing Oscillator Equations
References
10. Van der Pol–Duffing Oscillator Equation
10.1 Model Equation
10.2 Unforced Van der Pol–Duffing Oscillator Equation
10.3 Forced Van der Pol–Duffing Oscillator Equation
References
1.1 Introduction
1.2 Architecture of ANN
1.2.1 Feed-Forward Neural Network
1.2.2 Feedback Neural Network
1.3 Paradigms of Learning
1.3.1 Supervised Learning or Associative Learning
1.3.2 Unsupervised or Self-Organization Learning
1.4 Learning Rules or Learning Processes
1.4.1 Error Back-Propagation Learning Algorithm or Delta
Learning Rule
1.5 Activation Functions
1.5.1 Sigmoid Function
1.5.1.1 Unipolar Sigmoid Function
1.5.1.2 Bipolar Sigmoid Function
1.5.2 Tangent Hyperbolic Function
References
2. Preliminaries of Ordinary Differential Equations
2.1 Definitions
2.1.1 Order and Degree of DEs
2.1.2 Ordinary Differential Equation
2.1.3 Partial Differential Equation
2.1.4 Linear and Nonlinear Differential Equations
2.1.5 Initial Value Problem
2.1.6 Boundary Value Problem
References
3. Multilayer Artificial Neural Network
3.1 Structure of Multilayer ANN Model
3.2 Formulations and Learning Algorithm of Multilayer
ANN Model
3.2.1 General Formulation of ODEs Based on ANN Model
3.2.2 Formulation of nth-Order IVPs
3.2.2.1 Formulation of First-Order IVPs
3.2.2.2 Formulation of Second-Order IVPs
3.2.3 Formulation of BVPs
3.2.3.1 Formulation of Second-Order BVPs
3.2.3.2 Formulation of Fourth-Order BVPs
3.2.4 Formulation of a System of First-Order ODEs
3.2.5 Computation of Gradient of ODEs for Multilayer
ANN Model
3.3 First-Order Linear ODEs
3.4 Higher-Order ODEs
3.5 System of ODEs
References
4. Regression-Based ANN
4.1 Algorithm of RBNN Model
4.2 Structure of RBNN Model
4.3 Formulation and Learning Algorithm of RBNN Model
4.4 Computation of Gradient for RBNN Model
4.5 First-Order Linear ODEs
4.6 Higher-Order Linear ODEs
References
5. Single-Layer Functional Link Artificial Neural Network
5.1 Single-Layer FLANN Models
5.1.1 ChNN Model
5.1.1.1 Structure of the ChNN Model
5.1.1.2 Formulation of the ChNN Model
5.1.1.3 Gradient Computation of the ChNN Model
5.1.2 LeNN Model
5.1.2.1 Structure of the LeNN Model
5.1.2.2 Formulation of the LeNN Model
5.1.2.3 Gradient Computation of the LeNN Model
5.1.3 HeNN Model
5.1.3.1 Architecture of the HeNN Model
5.1.3.2 Formulation of the HeNN Model
5.1.4 Simple Orthogonal Polynomial–Based Neural
Network (SOPNN) Model
5.1.4.1 Structure of the SOPNN Model
5.1.4.2 Formulation of the SOPNN Model
5.1.4.3 Gradient Computation of the SOPNN Model
5.2 First-Order Linear ODEs
5.3 Higher-Order ODEs
5.4 System of ODEs
References
6. Single-Layer Functional Link Artificial Neural Network
with Regression-Based Weights
6.1 ChNN Model with Regression-Based Weights
6.1.1 Structure of the ChNN Model
6.1.2 Formulation and Gradient Computation
of the ChNN Model
6.2 First-Order Linear ODEs
6.3 Higher-Order ODEs
References
7. Lane–Emden Equations
7.1 Multilayer ANN-Based Solution of Lane–Emden Equations
7.2 FLANN-Based Solution of Lane–Emden Equations
7.2.1 Homogeneous Lane–Emden Equations
7.2.2 Nonhomogeneous Lane–Emden Equation
References
8. Emden–Fowler Equations
8.1 Multilayer ANN-Based Solution of Emden–Fowler
Equations
8.2 FLANN-Based Solution of Emden–Fowler Equations
References
9. Duffing Oscillator Equations
9.1 Governing Equation
9.2 Unforced Duffing Oscillator Equations
9.3 Forced Duffing Oscillator Equations
References
10. Van der Pol–Duffing Oscillator Equation
10.1 Model Equation
10.2 Unforced Van der Pol–Duffing Oscillator Equation
10.3 Forced Van der Pol–Duffing Oscillator Equation
References
Descriere
Differential equations play a vital role in the fields of engineering and science. Problems in engineering and science can be modeled using ordinary or partial differential equations. Analytical solutions of differential equations may not be obtained easily, so numerical methods have been developed to handle them.